Basis independent parametrisations of R parity violation in the soft SUSY breaking sector Sacha

The magnitude of R-parity violating coupling constants depends on which direction in the space of weak doublets with hypercharge = −2 corresponds to the Higgs. To address this “basis dependence”, one can construct combinations of coupling constants that are invariant under these basis transformations, and which parametrise how much R parity violation is present in the Lagrangian (analogous to Jarlskog invariants for CP violation). This has previously been done for the Higgs vev and the R parity violating couplings constants in the superpotential. In this letter, I build invariants that include soft SUSY breaking interactions, and briefly discuss their relation to invariants involving the Higgs vev. This completes the construction of invariants based on the MSSM with baryon parity. In the Standard Model, it is not possible to write down any renormalisable interactions that violate either baryon number (B) or lepton number (L)[1]. This is a consequence of the gauge symmetries and the particle content. In the supersymmetric extension of the Standard Model [2] there are many new particles, and it becomes possible to have renormalisable B or L non-conserving interactions. However, since neither B nor L violation has been observed in the laboratory, these interactions are often removed by imposing a symmetry. There are various symmetries that prevent the B or L violating renormalisable interactions in the supersymmetric Standard Model [3]. The most common is a discrete symmetry called R-parity, refered to as R in this letter. It alots each particle a multiplicative quantum number :(−1)2S+3B+L, where S is the particle spin. One can also allow the renormalisable B or L violating interactions to be present, but require the coupling constants to be sufficiently small to statisfy experimental bounds [4]. In this case, it is not desirable for the B and L violating interactions to be simultaneously present, because they mediate rapid proton decay. The bound on the product (lepton number violating Yukawa-type coupling)×(baryon number violating Yukawa-type coupling) varies from 10 to 10 [5], depending on the generation indices of the coupling constants. One therefore usually requires that either B or L be conserved. In this paper, I will assume that baryon number is exactly conserved in the renormalisable interactions. The superpotential for the Minimal Supersymmetric Standard Model (MSSM) with R-parity imposed is W = μH1H2 + h pq u H2QpU c q + h pq d H1QpD c q + h ij e H1LiE c j . (1) The Lagrangian also contains kinetic terms, gauge interactions, D-terms and soft SUSY breaking terms of the form soft masses +BHH1H2 + A pq u H2QpU c q + A pq d H1QpD c q + A ij e H1LiE c j . (2) I am abusively using capital letters for both superfields (as in eqn 1) and scalar component fields (as in eqn 2). Quark generation indices are p, q, r, s... and lepton indices are i, j, k.... Whether indices are up or down makes no difference. If instead of imposing R-parity, one merely requires that baryon number be conserved, there can be lepton number violating interactions in the superpotential: WL/ = ǫ H2Li + λ LiLjE c k + λ ′LiQpD c q (3) and in the soft terms: soft masses mixing L and H1 +B H2Li + A ijk λ LiLjE c k + A ipq λ LiQpD c q . (4) There are experimental upper bounds on these new couplings from various processes [4], such as Flavour-Changing-Neutral-Currents (FCNC), lepton flavour violation and lepton number nonconservation. However, some of these coupling constants can be made zero by a basis choice, so it is important to remember in which basis the bounds apply. In this letter I would like to approach this problem in a different way; I construct combinations of coupling constants that are “basis independent” and that parametrise the amount of R-parity violation present in the Lagrangian. They are zero if R, or equivalently lepton number, is conserved. A simple example of this approach is to take the superpotential of equations (1) and (3) with one quark and lepton generation. It appears to have two R violating interactions: ǫH2L and λ LQD. (There is no λLiLjE c k interaction because λ ijk is antisymmetric on the ij indices.) It is well known that one of these can be rotated into the other by mixing H1 and L. If H ′ 1 = μ √ μ + ǫ H1 + ǫ √ μ + ǫ L L = ǫ √ μ + ǫ H1 − μ √ μ + ǫ L , (5) then the Lagrangian expressed in terms of H ′ 1 and L ′ contains no H2L ′ term. One could instead dispose of the λLQD term. The coupling constant combination that is invariant under basis redefinitions inH1, L space, zero ifR parity is conserved, and non-zero if it is not is μλ −hdǫ = (ǫ, μ)∧(λ′, hd). “Basis-independent” parametrisations of R parity violation have previously been constructed from subsets of the parameters of the R parity non-conserving Minimal Supersymmetric Standard Model (MSSM). An invariant parmetrising the R violation due to the misalignment between the neutral vev and the superpotential μ-term was constructed in [6] and subsequently much discussed [7]. Invariants measuring the R parity violation between superpotential couplings, including the one discussed in the previous paragraph, were discussed in [8, 9]. The aim of this letter is to construct the “missing” invariants involving soft terms. The invariants in [6, 8] measure lepton number violation, whatever the lepton flavour; the singlet lepton family index is summed. I will here follow the approach of [9] and build invariants that parametrise lepton number violation in each family. Note that these invariants do not measure lepton flavour violation when lepton number is conserved. Invariants with lepton flavour indices are more numerous, but have the advantage that their relation to R violating coupling constants (which have lepton flavour indices) is more direct. In this letter I will first introduce some notation, review the geometric interpretation of the invariants, and then construct invariants parametrising the R parity violation amoung the soft terms, and between the soft terms and the superpotential terms. Finally I will briefly discuss the relation